Eccentric Angle : Ellipse

Q. The tangent at a point whose eccentric angle is ${60}^{\circ }$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b),$ meets the auxiliary circle at $L$ and $M$. If $LM$ subtends a right angle at the centre, then eccentricity of the ellipse is

1. $\frac{1}{\sqrt{7}}$
2. $\frac{2}{\sqrt{7}}$
3. $\frac{3}{\sqrt{7}}$
4. $\frac{1}{2}$

Q. $S$ is one focus of an standard horizontal ellipse and $P$ is any point on the ellipse. If the maximum and minimum values of $SP$ are $m$ and $n$ respectively, then the length of semi minor axis is

1. $AM$ of $m,n$
2. $GM$ of $m,n$
3. $HM$ of $m,n$
4. none of these

Q. If the chord through the points whose eccentric angles are $\theta$ and $\varphi$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ passes through a focus, then possible value(s) of $\tan \left(\frac{\theta }{2}\right)\tan \left(\frac{\varphi }{2}\right)$ is/are

1. $\frac{1}{9}$
2. $-9$
3. $-\frac{1}{9}$
4. $9$